On transmuted skewed Laplace distribution and its application

Introduction

The Laplace distribution also called the double exponential distribution, is the distribution of differences between two independent variants with identical exponential distribution [1] . Noted that Laplace distribution arises as tractable "lifetime" models in many areas, including life testing and telecommunications. The Laplace distribution has been found to be very useful in different aspects of human endeavors. The distribution lacks skew parameter. The inability of the Laplace distribution to account for skewness led to the development of the skewed Laplace distribution. Therefore, the Laplace distribution was extended by adding skewness parameter which is used to model the skewness of the data [2] noted that in practical situations in which some skewness are presents, the Skew Laplace distribution is more flexible to model real data. The Skewed Laplace distribution has been used in Economics, Engineering, Finance and Biology. Also, it has been used to describe the logarithm of particles, analyze bacterial sizes [3] . Skewed Laplace distribution was developed to accommodate both skewness and leptokurtic properties of data set.

Methods

The Existing Skewed Laplace Distribution

The following are the probability density function, f(x) and the cumulative density function, F(x) of Skewed Laplace distribution by [2] .

𝑓(𝑥, 𝜃, 𝜀, 𝜇) = { (

Where 𝜃 is the scale parameter, 𝜀 is the skewness parameter, and 𝜇 is the location parameter.

The transmutation method

The transmutation method was proposed by [4] as follows:

is the Cumulative Distribution Function (CDF) of the proposed distribution and  

x F is Cumulative Distribution Function (CDF) of the baseline distribution, a is the shape parameter; when a=0, the Cumulative Distribution Function (CDF) of the proposed distribution will be the same as that of the baseline distribution. The probability density function of the proposed distribution,(𝑥) , can be obtained from (3) by differentiating (3) with respect to the random variable x . Then;

But 𝐹 1 (𝑥) = 𝑓(𝑥), 𝑔(𝑥) 𝑛𝑜𝑤 𝑏𝑒𝑐𝑜𝑚𝑒𝑠

By putting (1) and ( 2) in ( 5), we have:

(6) is the Transmuted Skewed Laplace Distribution (TSLD).

Presentation of Data Set

Two real life datasets are used to demonstrate the numerical applications of the proposed distribution. The data sets are presented with the summary of their statistics.

Data set I:

This dataset represent the survival times of 121 patients with breast cancer obtained from a large hospital in a period from 1929 to 1938 [5] . This data set has recently been studied by [6] on survival analysis. Table 1 presents the data set. Source: [5] Data Set II: This data represent the tensile strength, measured in GPA, of 69 carbon fibers tested under tension at gauge lengths of 20mm, [7] experiment. This data set has recently been studied by [8] . Table 2 presents the data set. The summary of the data sets are obtained and presented in table 3 and4.

Table 2: Tensile Strength of Carbon Fibres

Results and Discussion

Results

Discussion of Results

Table 5 shows the parameter estimates of the models when fitted on data set I and the measure of fitness criteria values for comparison. The mode of comparison of the proposed distribution with the existing distribution is based on the Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC) and the Haman Quinn Information Criteria (HQIC). The proposed four parameter transmuted skewed Laplace distribution is compared with two existing distributions, first is the Skewed Laplace Distribution (SLD) of three parameters while the second is the Four Parameter Weibull Distribution (FPWD). Each of these distributions are fitted into the data set I and the estimate of the parameters; log-likelihood (LL), AIC, BIC and HQIC values were obtained. The objective of fitting the proposed four parameter distribution and compared with other distributions is to determine it flexibility in modeling real life situations. The results obtained show that the proposed four parameter TSLD having the least AIC, BIC and HQIC values has a better fit than the parent distribution (SLD) of less parameter which shows that the proposed model can be used effectively to model real life situations than the SLD. The competing model with the proposed model is the four parameter Weibull distribution. The four parameter Weibull distribution has the least AIC, BIC and HQIC values than the proposed four parameter TSLD which shows that the FPWD can be used to model this real life situation (the survival data used here) effectively than the TSLD. This result shows an impressive analysis about improving on an existing distribution. Once an existing distribution is improved upon by the addition of extra parameter, the new distribution becomes more flexible than the parent distribution. However, it becomes a competitive case when models of equal parameter are compared just like the FPWD has demonstrated better fitness than it counterpart (TSLD). Table 6 shows the parameter estimates of the models when fitted on data set II and the measure of fitness criteria values for comparison. Each of the distributions is fitted into the data set II and the estimate of the parameters, log-likelihood, AIC, BIC and HQIC values were obtained. The results obtained showed that the proposed four parameter TSLD having the least AIC, BIC and HQIC values has a better fit than the parent distribution (SLD) of less parameter which shows that the proposed model can be used effectively to model real life situations than the SLD. The competing model with the proposed model is the four parameter Weibull distribution. The four parameter Weibull distribution has the least AIC, BIC and HQIC values than the proposed four parameter TSLD which shows that the FPWD can be used to model this real life situation effectively than the TSLD.

The simulation approach for this study is the Monte Carlo simulations approach. The distribution used is the proposed Transmuted Skewed Laplace Distribution (TSLD). show that as the sample size is increasing, the bias and the mean square error values of the proposed distribution are reducing respectively. This shows the flexibility nature of the proposed transmuted skewed Laplace distribution to model real life situations of different nature. This is in line with the findings of [9,10,11,12,13] . All these researchers found that by extending or improving on the existing distributions that the new distributions perform better than their respective parent distributions in terms of fitness which is one of the major finding of this study.

Conclusion

This study expanded the existing skewed Laplace distribution by the injection of a shape parameter into the existing 3-parameter skewed Laplace distribution to make it a 4-parameter distribution. The transmuted skewed Laplace distribution was compared with existing skewed Laplace distribution using measures of fitness criteria such as Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC) and Haman Quinn Information Criteria (HQIC). Application of this distribution was demonstrated empirically using survival times of 121 patients with breast cancer obtained from a large hospitals in a period from 1929 to 1938 [5] and tensile strength of 69 carbon fibres tested under tension at gauge length of 20mm of the experiment conducted by [7] . Results were generated using R statistical software. The results obtained showed that the proposed four parameter Transmuted Skewed Laplace Distribution (TSLD) having the least AIC, BIC and HQIC values has better fit than the Skewed Laplace Distribution (SLD) of less parameter, which shows that the proposed model can be used effectively to model real life situation than the SLD. Also, the Four Parameter Weibull Distribution (FPWD) which is a competing model outperformed the Transmuted Skewed Laplace Distribution (TSLD).

References

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